Dantzig

Mathematical Optimization for Elixir — Write optimization problems naturally, like mathematical notation, with automatic linearization and powerful pattern-based modeling.

ℹ️ Two supported APIs: The DSL (Problem.define do…end) is ideal for static problems; the Functional API (Problem.new_variable, Problem.add_constraint, etc.) is the right choice for dynamic/runtime problems. Only Problem.maximize/2 and Problem.minimize/2 are deprecated. See Choosing an API Style and the Deprecation Notice.

require Dantzig.Problem, as: Problem

# Pattern-based N-dimensional modeling
problem =
  Problem.define do
    new(direction: :maximize)

    # Create x[i,j] for i=1..8, j=1..8 — that's 64 variables in one line!
    variables("x", [i <- 1..8, j <- 1..8], :binary, "Queen position")

    # One queen per row: sum over j for each i
    constraints([i <- 1..8], sum(x[i][:_]) == 1, "One queen per row")

    # One queen per column: sum over i for each j
    constraints([j <- 1..8], sum(x[:_][j]) == 1, "One queen per column")

    # Maximize queens placed (will be 8 for valid N-Queens solution)
    objective(sum(x[:_][:_]), direction: :maximize)
  end

{:ok, solution} = Dantzig.solve(problem)

✨ What Makes Dantzig Special

Pattern-Based Modeling: Create N-dimensional variables with intuitive generator syntax

# Instead of manually creating x11, x12, x13, x21, x22, x23, ...
variables("x", [i <- 1..3, j <- 1..3], :binary)

# Use natural patterns: x[i][:_] sums over second dimension
constraints([i <- 1..3], sum(x[i][:_]) == 1, "Row constraint")

Automatic Linearization: Non-linear expressions become linear constraints automatically

# These work out of the box — no manual linearization needed!
constraints(abs(x) + max(x, y, z) <= 5, "Non-linear with auto-linearization")
constraints(a AND b AND c, "Logical AND constraint")

Model Parameters: Pass runtime data directly into your optimization models

# Use a map with integer keys matching the generator range
# Bracket notation is used uniformly: costs[i] for constants, x[i] for variables
params = %{costs: %{1 => 10, 2 => 20, 3 => 30}, capacity: 100}

problem = Problem.define(model_parameters: params) do
  variables("x", [i <- 1..3], :integer, min_bound: 0)

  constraints(sum(for i <- 1..3, do: costs[i] * x[i]) <= capacity, "Budget")
  objective(sum(for i <- 1..3, do: x[i]), :maximize)
end

Problem.modify: Build problems incrementally

# Start with base problem
base = Problem.define do
  variables("x", [i <- 1..3], :continuous)
  constraints([i <- 1..3], x[i] >= 0)
end

# Add more constraints and variables
problem = Problem.modify(base) do
  variables("y", [j <- 1..2], :binary)
  constraints(x[1] + x[2] + x[3] <= 10, "Capacity")
  objective(x[1] + 2*x[2] + 3*x[3], :maximize)
end

Multiple Modeling Styles: Choose the approach that fits your problem

• Simple syntax for basic problems
• Pattern-based for N-dimensional problems
• Model parameters for configurable problems
• Problem.modify for incremental building
• Explicit control when you need it
• AST transformations for advanced use cases

🚀 Quick Start

Installation

Add to your mix.exs:

def deps do
  [
    {:dantzig, "~> 0.2.0"}
  ]
end

Your First Problem

require Dantzig.Problem, as: Problem

problem =
  Problem.define do
    new(direction: :maximize)

    variables("x", :continuous, min_bound: 0, description: "Items to produce")
    variables("y", :continuous, min_bound: 0, description: "Items to sell")

    constraints(x + 2*y <= 14, "Resource constraint")
    constraints(3*x - y <= 0, "Quality constraint")

    objective(3*x + 4*y, direction: :maximize)
  end

{:ok, solution} = Dantzig.solve(problem)
IO.inspect({solution.objective, solution.variables})

📖 Documentation & Learning

Level	Guide	Description
🏃 Beginner	Quick Start	Your first optimization problem
📚 Intermediate	Tutorial	Complete guide with examples
🏗️ Advanced	Architecture	System design deep dive
🔧 Reference	API Docs	Complete function reference

Generate full docs locally:

mix docs

🎯 Core Features

Pattern-Based Variables

Create complex variable structures with simple generators:

# 2D transportation problem: supply[i] to demand[j]
variables("transport", [i <- 1..3, j <- 1..4], :continuous, min_bound: 0)

# 3D production planning: product[p] in period[t] at plant[f]
variables("produce", [p <- products, t <- 1..12, f <- facilities], :integer, min_bound: 0)

# 4D chess tournament: player[a] vs player[b] in round[r] at table[t]
variables("game", [a <- 1..8, b <- 1..8, r <- 1..7, t <- 1..4], :binary)

Automatic Linearization

Non-linear expressions become linear constraints behind the scenes:

# Absolute values
constraints(abs(inventory) <= max_inventory, "Absolute inventory")

# Maximum functions
constraints(max(profit1, profit2, profit3) >= target, "Max profit")

# Minimum functions
constraints(min(cost1, cost2) <= budget, "Min cost")

# Logical operations (binary variables)
constraints(decision1 AND decision2 AND decision3, "All decisions required")
constraints(decision1 OR decision2 OR decision3, "At least one decision")

Flexible Constraint Patterns

Express complex constraints naturally:

# Sum over specific dimensions
constraints([i <- 1..5], sum(x[i][:_]) == 1, "One per row")
constraints([j <- 1..5], sum(x[:_][j]) == 1, "One per column")

# Complex aggregations
constraints([i <- 1..3], sum(x[i][:_]) >= demand[i], "Demand satisfaction")

# Multi-dimensional constraints
constraints([i <- 1..3, j <- 1..3], x[i][j] <= capacity[i][j], "Capacity limits")

💡 Examples

Complete, runnable examples are available in the Examples Directory:

Simple Problems

• Resource Allocation: mix run docs/user/examples/simple_working_example.exs
• Knapsack: mix run docs/user/examples/knapsack_problem.exs
• Assignment: mix run docs/user/examples/assignment_problem.exs

Medium Problems

• Transportation: mix run docs/user/examples/transportation_problem.exs
• Production Planning: mix run docs/user/examples/production_planning.exs
• Blending: mix run docs/user/examples/blending_problem.exs

Complex Problems

• N-Queens: mix run docs/user/examples/nqueens_dsl.exs
• Pattern Operations: mix run docs/user/examples/pattern_based_operations_example.exs
• Network Flow: mix run docs/user/examples/network_flow.exs

See the Examples Directory for a complete categorized list of all examples.

🏗️ Choosing an API Style

Dantzig exposes two current, supported APIs and one deprecated one. Understanding the difference helps you pick the right tool.

API 1: The DSL (Problem.define do … end)

A compile-time macro that lets you write optimization problems in a declarative, mathematical style. Internally it expands to the Functional API (below) via DSLReducer.

require Dantzig.Problem, as: Problem

problem = Problem.define do
  new(direction: :maximize)
  variables("x", [i <- 1..3], :continuous, min_bound: 0)
  constraints([i <- 1..3], x[i] <= 10, "Bound")
  objective(sum(x[:_]), direction: :maximize)
end

Use the DSL when:

• The problem structure is statically known at compile time
• You want concise, mathematical notation
• You're building textbook-style problems (knapsack, N-queens, blending, etc.)

Limitation: outer-scope Elixir variables referenced inside max_bound:, description:, or string-interpolated constraint names may not resolve correctly — they are captured as AST, not evaluated values. Use model_parameters: to pass runtime data into the DSL:

Problem.define(model_parameters: %{capacity: 100}) do
  new(direction: :minimize)
  variables("x", [i <- 1..3], :continuous, min_bound: 0)
  constraints(sum(x[:_]) <= capacity, "Capacity")   # capacity resolved at runtime ✓
  objective(sum(x[:_]), direction: :minimize)
end

---

API 2: The Functional API

Regular Elixir functions that Problem.define expands to. Fully public, supported, and the right choice for dynamic problems where the shape (number of variables, constraints) is determined at runtime — for example, from database records or user input.

Key functions:

alias Dantzig.{Polynomial, Constraint}
require Dantzig.Constraint, as: Constraint

problem = Problem.new(direction: :maximize)

# Add variables one at a time; each returns {updated_problem, polynomial_handle}
{problem, x} = Problem.new_variable(problem, "x", type: :continuous, min_bound: 0)
{problem, y} = Problem.new_variable(problem, "y", type: :continuous, min_bound: 0)

# Build the objective incrementally
problem = problem
  |> Problem.increment_objective(Polynomial.multiply(x, 3))
  |> Problem.increment_objective(Polynomial.multiply(y, 4))

# Add constraints (Constraint.new_linear is a macro — require it)
problem = Problem.add_constraint(problem,
  Constraint.new_linear(x + 2*y <= 14, name: "resource"))

{:ok, solution} = Dantzig.solve(problem)

Use the Functional API when:

• The problem shape depends on runtime data (e.g. from a database or API)
• You're building a helper that programmatically generates variables or constraints in a loop
• You need fine-grained control over variable naming

---

❌ Deprecated: Old Objective API

The following functions still exist but are deprecated and will be removed in v1.0.0:

# ❌ DEPRECATED — sets objective but discards any previously accumulated terms
problem = Problem.maximize(problem, 3*x + 4*y)
problem = Problem.minimize(problem, cost)

Replace with:

# ✅ Set direction in new/1, build objective with increment_objective/2
problem = Problem.new(direction: :maximize)
problem = Problem.increment_objective(problem, Polynomial.multiply(x, 3))
problem = Problem.increment_objective(problem, Polynomial.multiply(y, 4))

# or set it all at once with set_objective/2 + direction in new/1
problem = Problem.set_objective(problem, 3*x + 4*y)

The old add_variables/5 macro (with explicit problem as first arg) is similarly deprecated in favour of variables/5 inside a define block or Problem.new_variable/3 in the Functional API.

World-Class Optimization Power: Dantzig integrates seamlessly with the HiGHS solver, providing access to cutting-edge optimization algorithms.

Automatic Binary Management: The HiGHS solver binary is automatically downloaded and managed for your platform.

High Performance: Leverages HiGHS's advanced algorithms for linear programming, mixed-integer programming, and quadratic programming.

# Automatic HiGHS integration - no manual setup required!
{:ok, solution} = Dantzig.solve(problem)

# Custom configuration (optional)
config :dantzig, :highs_binary_path, "/usr/local/bin/highs"
config :dantzig, :highs_version, "1.9.0"

# Advanced solver options
config :dantzig, :solver_options, [
  parallel: true,
  presolve: :on,
  time_limit: 300.0
]

🔧 Configuration

Dantzig automatically downloads the HiGHS solver binary for your platform:

# Custom binary path (optional)
config :dantzig, :highs_binary_path, "/usr/local/bin/highs"

# HiGHS version (default: "1.9.0")
config :dantzig, :highs_version, "1.9.0"

📊 Current Capabilities

Feature	Status	Notes
Linear Programming	✅ Complete	Full support
Quadratic Programming	✅ Complete	Degree ≤ 2 expressions
Pattern-based Modeling	✅ Complete	N-dimensional variables
Automatic Linearization	✅ Complete	abs, max/min, logical ops
Mixed-Integer Variables	⚠️ Tracked	Types defined, LP serialization pending
Custom Operators	🚧 Reserved	:in operator for future use

🤝 Contributing

We welcome contributions! Explore the architecture docs to understand the system design, and feel free to submit issues and pull requests on GitHub.

Key Areas for Contribution:

• Mixed-integer LP serialization
• Additional non-linear function support
• Performance optimizations
• Documentation and examples

📜 License

MIT License - see LICENSE.TXT for details.

🙏 Acknowledgments

• HiGHS - World-class optimization solver
• JuliaBinaryWrappers - Pre-compiled binaries
• Elixir Community - Inspiration and valuable feedback

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Ready to optimize? Start with the Quick Start Guide or dive into the Tutorial for comprehensive examples!